Abstract

Let Fn(x) and Gn(x) be the empirical distribution functions of two
independent samples, each of size n, in the case where the elements of the
samples are independent random variables, each having the same continuous
distribution function V(x) over the interval (0,1). Define a statistic θn by
θn/n=∫01[Fn(x)−Gn(x)]dV(x)−min0≤x≤1[Fn(x)−Gn(x)].
In this paper the limits of E{(θn/2n)r}(r=0,1,2,…) and P{θn/2n≤x}
are determined for n→∞. The problem of finding the asymptotic behavior of
the moments and the distribution of θn as n→∞ has arisen in a study of the
fluctuations of the inventory of locomotives in a randomly chosen railway
depot.

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